Phase modulation in single-mode optical fibers has been successfully used as a transduction mechanism for detecting environmental changes such as pressure, acoustic fields, magnetic and electric fields, and temperature. Mach-Zehnder interferometric fiber-optic sensors have been studied analytically and have been tested both in the laboratory and in real environments and demonstrate very promising performance.
With an interferometric fiber optic sensor as shown in FIG. 1, optimum performance is achieved with maximum signal to noise ratio which signal is obtained by maximizing the sensitivity of the sensing fiber and desensitizing the sensitivity of the lead and reference fibers. U.S. Pat. No. 4,427,263 issued to Lagakos et al. on Jan. 24, 1984, discloses pressure insensitive optical fibers for use in the reference and lead arms of the interferometer.
The acoustic sensitivity of an optical fiber can be maximized by appropriately selecting the fiber composition and design, as will be disclosed.
The pressure sensitivity of the optical phase in a fiber is defined as the magnitude of .DELTA..phi./.phi..DELTA.P, where .DELTA..phi. is the shift in the phase delay .phi. due to a pressure change .DELTA.P. If a given pressure change, .DELTA.P, results in a fiber core axial strain .epsilon..sub.z and radial strain .epsilon..sub.r, then it can be shown that: ##EQU1## Here P.sub.11 and P.sub.12 are the elasto-optic coefficients of the core and n is the refractive index of the core. The first term in equation (1) is the part of .DELTA..phi./.phi..DELTA.P which is due to the fiber length change, while the second and third terms are the parts due to the refractive index modulation of the core, which is related to the photoelastic effect. Stated another way .DELTA..phi./.phi..DELTA.P=the algebraic sum of the phase change due to the fiber length change plus phase change due to refractive index change. The objective is to maximize the fiber sensitivity .DELTA..phi./.phi..DELTA.P.
A typical optical fiber (FIG. 2a) is composed of a core, cladding, and a substrate from glasses having similar properties. This glass fiber is usually coated with a soft rubber and then with a hard plastic. In order to calculate the sensitivity as given in Eq.(1) the strains in the core .epsilon..sub.z and .epsilon..sub.r must be related to properties of the various layer of the fiber. The strains .epsilon..sub.z and .epsilon..sub.r, which are related to the geometry and composition of the fiber, can be calculated from the stresses and displacements in the various fiber layers by applying the appropriate boundary conditions. In this analysis we have taken into account the exact geometry of a typical four layer fiber as shown in FIG. 2a.
The polar stresses .tau..sub.r, .tau..sub..theta., and .tau..sub.z in the fiber are related to the strains .epsilon..sub.r, .epsilon..sub..theta., and .epsilon..sub.z as follows: ##EQU2## where i is the layer index (0 for the core, 1 for the cladding, etc.) and .lambda..sup.i and .mu..sup.i are the Lame parameters which are related to the Young's modulus, E.sup.i, and Poisson's ratio, V.sup.i, as follows: ##EQU3##
For a cylinder the strains can be obtained from the Lame solutions ##EQU4## where U.sub.o.sup.i, U.sub.l.sup.i, and W.sub.o.sup.i are constants to be determined. Since the strains must be finite at the center of the core, U.sub.l.sup.o =0.
For a fiber with m layers, the constants U.sub.o.sup.i, U.sub.l.sup.i, and W.sub.o.sup.i in Eq. (4) are determined from the boundary conditions: EQU .tau..sub.r.sup.i .vertline..sub.r=ri =.tau..sub.r.sup.i+1 .vertline..sub.r=ri,(i=0,1, . . . . , m-1) (5) EQU .mu..sub.r.sup.i .vertline..sub.r=ri =.mu..sub.r.sup.i+1 .vertline..sub.r=ri,(i=0,1, . . . . , m-1) (6) EQU .tau..sub.r.sup.m .vertline..sub.r=rm =-P, (7) ##EQU5## EQU .epsilon..sub.z.sup.o =.epsilon..sub.z.sup.1 =. . . =.epsilon..sub.z.sup.m( 9)
where u.sub.r.sup.j (=.intg..epsilon..sub.r.sup.i dr) is the radial displacement in the i.sup.th layer, and r.sub.i and A.sub.i are the radius and the cross section area of the i.sup.th layer respectively. Equations (5) and (6) describe the radial stress and displacement continuity across the boundaries of the layers. Equations (7) and (8) assume that the applied pressure is hydrostatic. Equation (9) is the plane strain approximation which ignores end effects. For long thin cylinders, such as fibers, it introduces an error of less than 1%. Using the boundary conditions described by Eqs. (5)-(9), the constants U.sub.o.sup.i, U.sub.l.sup.i, and W.sub.o.sup.i are determined and .epsilon..sub.r.sup.o and .epsilon..sub.z.sup.o are calculated from Eq.(4). Then, from Eq.(1) the sensitivity, .DELTA..phi./.DELTA..phi.P, can be found.
In the past no one has recognized the relationship between the bulk modulus, Young's modulus of the fiber coating, the fiber thickness, and the the resultant acoustic pressure sensitivity of the optical fiber.